Quadrilateral Word Problems Study Guide (page 2)
Introduction to Quadrilateral Word Problems
[G]eometry is not true, it is advantageous.
—HENRI POINCARÉ (1854–1912)
This lesson will review the properties of quadrilaterals and the special types of quadrilaterals. Word problems involving quadrilaterals will be solved.
Quadrilaterals are four-sided polygons. This means that they are closed figures with four line segments as sides. The interior angles of every quadrilateral total 360°.
There are many types of special quadrilaterals. These quadrilaterals are classified, or named, based on their properties.
The first group of special quadrilaterals is the parallelograms.
A parallelogram is a quadrilateral with opposite sides congruent, or the same measure. Parallelograms also have opposite angles congruent, and the diagonals bisect each other. In addition, consecutive angles of any parallelogram are supplementary. In other words, angles that are next to each other, like angle A and angle B in the following figure, add to 180 degrees. This property and others are shown in the examples below.
A rhombus is a parallelogram with all four sides congruent. It could look like a square that is leaning over. Keep in mind that all squares are also rhombuses. Rhombuses have all the properties of parallelograms, in addition to the fact that the diagonals are perpendicular, or meet at right angles. Examples of rhombuses are shown in the following figures.
A rectangle is a parallelogram with four right angles. It could look like a parallelogram standing up straight. Rectangles also have all of the properties of parallelograms, in addition to the fact that the diagonals are congruent. Following are two examples of rectangles.
A square is a rhombus with right angles. Therefore, all sides are congruent, and all angles are right angles. Squares have all of the properties of rhombuses, in addition to having four congruent angles and congruent diagonals. A square is shown next.
A square can be remembered as a rectangle with four congruent sides, or a rhombus with four right angles.
The second group of special quadrilaterals is the trapezoids.
A trapezoid is a quadrilateral with exactly one pair of parallel sides. Trapezoids have two parallel sides that are not the same measure; these sides are called the bases of the trapezoid. The two sides that are not parallel are called the legs of the trapezoid. Trapezoid examples are shown in the following figures.
An isosceles trapezoid, like an isosceles triangle, has two sides congruent. In this type of trapezoid, the legs are the same measure. Isosceles triangles also have congruent base angles and congruent diagonals. These properties are shown in the following figure.
All parallelograms have two pairs of parallel sides.
Quadrilateral Word Problems
Each example problem that follows uses the steps to solving word problems and the properties of quadrilaterals. Use these problems as a guide to solving quadrilateral word problems.
An isosceles trapezoid has base angles of 120°. What is the measure in each of the other angles of the trapezoid?
Read and understand the question. This question is looking for the measure of each of the unknown angles in an isosceles trapezoid. The measures of the base angles are given.
Make a plan. Use the problem solving strategy of drawing a picture to help with this question. In an isosceles trapezoid, the two legs are congruent and the base angles are congruent. The following figure represents this trapezoid.
Carry out the plan. The base angles of the trapezoid are congruent, so each of their measures is 120°. To find the measure of the other angles, subtract the sum of the base angles from 360° and divide by 2: 360 – 240 = 120, = 60° in each of the other angles. Thus, the angles measure 120°, 120°, 60°, and 60°, respectively.
Check your answer. To check this answer, add the measures of the four angles to be sure that the total number of degrees is 360: 120 + 120 + 60 + 60 = 360°, so this answer is checking.
A parallelogram has two opposite sides labeled x + 5 units and 2x – 3 units, respectively. What is the length of these opposite sides?
Read and understand the question. This question is looking for the length of each of the opposite sides in a parallelogram.
Make a plan. The lengths of opposite sides of a parallelogram are equal. Set the given expressions equal to each other, and solve for x. Then, substitute the value of x into one of the expressions to find the length of the sides.
Carry out the plan. Set the expressions equal to each other: x + 5 = 2x– 3. Subtract x from each side of the equation to get 5 = x – 3. Add 3 to each side of the equation to get 8 = x. Substitute x = 8 into the expression x + 5. 8 + 5 = 13. The length of each of the opposite sides is 13 units.
Check your answer. To check this answer, substitute x = 8 into the expression 2x – 3 to be sure the value is also 13.
- 2(8) – 3 = 16 – 3 = 13
This solution is checking.
Find practice problems and solutions for these concepts at Quadrilateral Word Problems Practice Questions.
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